Transverse isotropy

Transverse Isotropy is observed in sedimentary rocks at long wavelengths. Each layer has approximately the same properties in-plane but different properties through-the-thickness. The plane of each layer is the plane of isotropy and the vertical axis is the axis of symmetry.

A transversely isotropic material is one with physical properties that are symmetric about an axis that is normal to a plane of isotropy. This transverse plane has infinite planes of symmetry and thus, within this plane, the material properties are the same in all directions. Hence, such materials are also known as "polar anisotropic" materials.

This type of material exhibits hexagonal symmetry (though technically this ceases to be true for tensors of rank 6 and higher), so the number of independent constants in the (fourth-rank) elasticity tensor are reduced to 5 (from a total of 21 independent constants in the case of a fully anisotropicsolid). The (second-rank) tensors of electrical resistivity, permeability, etc. have two independent constants.

An example of a transversely isotropic material is the so-called on-axis unidirectional fiber composite lamina where the fibers are circular in cross section. In a unidirectional composite, the plane normal to the fiber direction can be considered as the isotropic plane, at long wavelengths (low frequencies) of excitation. In the figure to the right, the fibers would be aligned with the x2{\displaystyle x_{2}} axis, which is normal to the plane of isotropy.

In terms of effective properties, geological layers of rocks are often interpreted as being transversely isotropic. Calculating the effective elastic properties of such layers in petrology has been coined Backus upscaling, which is described below.

The material matrix K__{\displaystyle {\underline {\underline {\boldsymbol {K}}}}} has a symmetry with respect to a given orthogonal transformation (A{\displaystyle {\boldsymbol {A}}}) if it does not change when subjected to that transformation.
For invariance of the material properties under such a transformation we require

Using θ=π{\displaystyle \theta =\pi } in the A__{\displaystyle {\underline {\underline {\boldsymbol {A}}}}} matrix implies that K13=K31=K23=K32=0{\displaystyle K_{13}=K_{31}=K_{23}=K_{32}=0}. Using θ=π2{\displaystyle \theta ={\tfrac {\pi }{2}}} leads to K11=K22{\displaystyle K_{11}=K_{22}} and K12=−K21{\displaystyle K_{12}=-K_{21}}. Energy restrictions usually require K12,K21≥0{\displaystyle K_{12},K_{21}\geq 0} and hence we must have K12=K21=0{\displaystyle K_{12}=K_{21}=0}. Therefore, the material properties of a transversely isotropic material are described by the matrix

Using the specific values of θ{\displaystyle \theta } in matrix A__{\displaystyle {\underline {\underline {\boldsymbol {A}}}}},[3] it can be shown that the fourth-rank elasticity stiffness tensor may be written in 2-index Voigt notation as the matrix

The elasticity stiffness matrix Cij{\displaystyle C_{ij}} has 5 independent constants, which are related to well known engineering elastic moduli in the following way. These engineering moduli are experimentally determined.

In geophysics, a common assumption is that the rock formations of the crust are locally polar anisotropic (transversely isotropic); this is the simplest case of geophysical interest. Backus upscaling[4] is often used to determine the effective transversely isotropic elastic constants of layered media for long wavelength seismic waves.

Assumptions that are made in the Backus approximation are:

All materials are linearly elastic

No sources of intrinsic energy dissipation (e.g. friction)

Valid in the infinite wavelength limit, hence good results only if layer thickness is much smaller than wavelength

The statistics of distribution of layer elastic properties are stationary, i.e., there is no correlated trend in these properties.

For shorter wavelengths, the behavior of seismic waves is described using the superposition of plane waves. Transversely isotropic media support three types of elastic plane waves:

A layered model of homogeneous and isotropic material, can be up-scaled to a transverse isotropic medium, proposed by Backus.[4]

Backus presented an equivalent medium theory, a heterogeneous medium can be replaced by a homogeneous one that predicts wave propagation in the actual medium.[5] Backus showed that layering on a scale much finer than the wavelength has an impact and that a number of isotropic layers can be replaced by a homogeneous transversely isotropic medium that behaves exactly in the same manner as the actual medium under static load in the infinite wavelength limit.

If each layer i{\displaystyle i} is described by 5 transversely isotropic parameters (ai,bi,ci,di,ei){\displaystyle (a_{i},b_{i},c_{i},d_{i},e_{i})}, specifying the matrix

Solutions to wave propagation problems in linear elastic transversely isotropic media can be constructed by superposing solutions for the quasi-P wave, the quasi S-wave, and a S-wave polarized orthogonal to the quasi S-wave.
However, the equations for the angular variation of velocity are algebraically complex and the plane-wave velocities are functions of the propagation angle θ{\displaystyle \theta } are.[6] The direction dependent wave speeds for elastic waves through the material can be found by using the Christoffel equation and are given by[7]

where θ{\displaystyle {\begin{aligned}\theta \end{aligned}}} is the angle between the axis of symmetry and the wave propagation direction, ρ{\displaystyle \rho } is mass density and the Cij{\displaystyle C_{ij}} are elements of the elastic stiffness matrix. The Thomsen parameters are used to simplify these expressions and make them easier to understand.

Thomsen parameters[8] are dimensionless combinations of elastic moduli that characterize transversely isotropic materials, which are encountered, for example, in geophysics. In terms of the components of the elastic stiffness matrix, these parameters are defined as:

where index 3 indicates the axis of symmetry (e3{\displaystyle \mathbf {e} _{3}}) . These parameters, in conjunction with the associated P wave and S wave velocities, can be used to characterize wave propagation through weakly anisotropic, layered media. Empirically, the Thomsen parameters for most layered rock formations are much lower than 1.

The name refers to Leon Thomsen, professor of geophysics at the University of Houston, who proposed these parameters in his 1986 paper "Weak Elastic Anisotropy".

In geophysics the anisotropy in elastic properties is usually weak, in which case δ,γ,ϵ≪1{\displaystyle \delta ,\gamma ,\epsilon \ll 1}. When the exact expressions for the wave velocities above are linearized in these small quantities, they simplify to

are the P and S wave velocities in the direction of the axis of symmetry (e3{\displaystyle \mathbf {e} _{3}}) (in geophysics, this is usually, but not always, the vertical direction). Note that δ{\displaystyle \delta } may be further linearized, but this does not lead to further simplification.

The approximate expressions for the wave velocities are simple enough to be physically interpreted, and sufficiently accurate for most geophysical applications. These expressions are also useful in some contexts where the anisotropy is not weak.

^We can use the values θ=π{\displaystyle \theta =\pi } and θ=π2{\displaystyle \theta ={\tfrac {\pi }{2}}} for a derivation of the stiffness matrix for transversely isotropic materials. Specific values are chosen to make the calculation easier.